Manuscript III-B@1 · III-B-v1.tex · proposition · d2c98f62

Proposition: The action preserves the retained equations

The retained first-order jet and its retarded section › The equation space and its action · lines 972–993 · proof included or linked

III-B@1#prop:jet-action

Equation (eq:jet-action) defines an additive action on $J_{R}$ :

T_{ξ} T_{ζ} = T_{ξ + ζ}, T_{ξ}^{- 1} = T_{- ξ} .

Proof. The support and past-trivial conditions are preserved. The scalar equation transforms to

P (ϕ_{1} + X_{ξ} ϕ_{0}) + L_{h + K ξ} ϕ_{0} = P ϕ_{1} + L_{h} ϕ_{0} + P X_{ξ} ϕ_{0} - [P, X_{ξ}] ϕ_{0} = X_{ξ} P ϕ_{0} = 0.

The metric equation follows from $G^{(1)} K = 0$ . The group identities are immediate.

Displayed formulas in this passage

unlabeled · line 985
$P (ϕ_{1} + X_{ξ} ϕ_{0}) + L_{h + K ξ} ϕ_{0} = P ϕ_{1} + L_{h} ϕ_{0} + P X_{ξ} ϕ_{0} - [P, X_{ξ}] ϕ_{0} = X_{ξ} P ϕ_{0} = 0.$
unlabeled · line 976
$T_{ξ} T_{ζ} = T_{ξ + ζ}, T_{ξ}^{- 1} = T_{- ξ} .$